It's right that a well organised team wouldn't look favourably on sights that didn't give 4 clicks to the minute (whether because of the sight base or the design of sight). In practice that isn't really an issue in the UK because 30" barrles are very standard (though a few longer ones are appearing) and modern(ish) rear sights are all quarter minute clicks.
I've never heard of anyone not being allowed on a team because they had a particular make of rifle action. Ultimately it's all about scores (or, for team shooting, vertical groups) and if your kit doesn't allow you to produce the goods then that would hinder your chances. However, if you can shoot good groups then nobody would care what you do it with as long as it's mechanically reliable.
Here are my tips on fullbore rifles:
The make of action isn't really important for accuracy. Any of the actions mentioned in this thread will do the job and the comments about ease of operation and availability of parts are worth considering.
The barrel is critical to accuracy. This is a problem when buying second hand because you may not know if the rifle shoots well and, if it does, how long it will continue to do that. Hopefully the seller's scores might give you an idea of whether it's any good but somehow you need to try it out, or get someone you trust to try it out for you. The only consolation I can offer here is that even if you buy a new barrel at great expense you still don't know if it will be any good!
The stock and the bedding of the action is also important but a good visual check can help with those.
Don't necessarily be seduced by very fancy adjustable stocks. If you look at the GB team, they will almost all use simple wooden stocks probably with an adjustable (up and down) buttplate and, perhaps, an adjustable cheek piece. Firing points are never the same and you can never adjust a stock so that it will feel the same every time. Getting used to the variation in position is a valuable skill for fullbore shooting.